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The integral
 
The integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470301.png" /></td> </tr></table>
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$$
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f ( n)  = H \{ F ( x) \}  = \int\limits _ {- \infty } ^  \infty 
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e ^ {- x  ^ {2} } H _ {n} ( x) F ( x)  d x ,\ \
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n = 0 , 1 \dots
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$$
 +
 
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where  $  H _ {n} ( x) $
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are the [[Hermite polynomials|Hermite polynomials]]. The inversion formula is
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470302.png" /> are the [[Hermite polynomials|Hermite polynomials]]. The inversion formula is
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$$
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F ( x)  = \sum _ { n= 0} ^  \infty 
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\frac{1}{\sqrt \pi }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470303.png" /></td> </tr></table>
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\frac{f ( n) }{2  ^ {n} n ! }
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H _ {n} ( x)  = \
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H  ^ {-1} \{ f ( n) \} ,\  - \infty < x < \infty ,
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$$
  
 
provided that the series converges. The Hermite transform reduces the operator
 
provided that the series converges. The Hermite transform reduces the operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470304.png" /></td> </tr></table>
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$$
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R [ F ( x) ]  = e ^ {x  ^ {2} }
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\frac{d}{dx}
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 +
\left [ e ^ {x  ^ {2} }
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\frac{d}{dx}
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F ( x) \right ]
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$$
  
 
to an algebraic one by the formula
 
to an algebraic one by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470305.png" /></td> </tr></table>
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$$
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H \{ R [ F ( x) ] \}  = - 2 n f ( n) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470306.png" /> and all its derivatives up to and including the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470307.png" />-th order are bounded, then
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If $  F $
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and all its derivatives up to and including the $  p $-
 +
th order are bounded, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470308.png" /></td> </tr></table>
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$$
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H \{ F ^ { ( p) } ( x) \}  = f ( n + p ) .
 +
$$
  
The Hermite transform has also been introduced for a special class of generalized functions (see [[#References|[2]]]). They are used to solve differential equations containing the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047030/h0470309.png" />.
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The Hermite transform has also been introduced for a special class of generalized functions (see [[#References|[2]]]). They are used to solve differential equations containing the operator $  R $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Debnath,  "On the Hermite transform"  ''Mat. Vesnik'' , '''1'''  (1964)  pp. 285–292</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transforms" , Wiley  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Debnath,  "On the Hermite transform"  ''Mat. Vesnik'' , '''1'''  (1964)  pp. 285–292</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transforms" , Wiley  (1968)</TD></TR></table>

Latest revision as of 21:42, 27 December 2020


The integral

$$ f ( n) = H \{ F ( x) \} = \int\limits _ {- \infty } ^ \infty e ^ {- x ^ {2} } H _ {n} ( x) F ( x) d x ,\ \ n = 0 , 1 \dots $$

where $ H _ {n} ( x) $ are the Hermite polynomials. The inversion formula is

$$ F ( x) = \sum _ { n= 0} ^ \infty \frac{1}{\sqrt \pi } \frac{f ( n) }{2 ^ {n} n ! } H _ {n} ( x) = \ H ^ {-1} \{ f ( n) \} ,\ - \infty < x < \infty , $$

provided that the series converges. The Hermite transform reduces the operator

$$ R [ F ( x) ] = e ^ {x ^ {2} } \frac{d}{dx} \left [ e ^ {x ^ {2} } \frac{d}{dx} F ( x) \right ] $$

to an algebraic one by the formula

$$ H \{ R [ F ( x) ] \} = - 2 n f ( n) . $$

If $ F $ and all its derivatives up to and including the $ p $- th order are bounded, then

$$ H \{ F ^ { ( p) } ( x) \} = f ( n + p ) . $$

The Hermite transform has also been introduced for a special class of generalized functions (see [2]). They are used to solve differential equations containing the operator $ R $.

References

[1] L. Debnath, "On the Hermite transform" Mat. Vesnik , 1 (1964) pp. 285–292
[2] A.G. Zemanian, "Generalized integral transforms" , Wiley (1968)
How to Cite This Entry:
Hermite transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_transform&oldid=13762
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article